Elusive Heat, Chaotic Patterns

For decades scientists have used experiments replicating a classic form of heat transfer known as Rayleigh-Benard convection to study the effects of a warmer substance displacing a cooler one. These tightly controlled investigations of fluid dynamics allow scientists to scrutinize the fundamentals of behavior that is ubiquitous but otherwise elusive for study. A simmering pot of water, home heating and cooling, a weather system, or the fantastic furnace that is the sun -- all possess convection attributes.

For the past several years, James D. Gunton, Haowen Xi and Jorge Vinals have used Pittsburgh Supercomputing Center's CRAY Y-MP and C90 to simulate Rayleigh-Benard convection. The Gunton team's simulations have revealed unpredicted chaotic patterns in R-B convection, unveiled new R-B convection structures and paved the way for mathematically accurate, three-dimensional renditions of chaotic behavior.

"We're trying to understand pattern formation and chaotic behavior in nature," says Gunton. "As a result of using supercomputing to simulate R-B convection, we're discovering previously unknown chaos in fluids."

Dependable Rayleigh-Benard

These images show the surface topography of parallel roll (top) and hexagon structures (bottom) in Rayleigh-Benard convection. Color corresponds to temperature, increasing from violet to red.

Rayleigh-Benard convection is significant because for a comparatively easy-to-replicate phenomenon, it continues to provide insight into understanding how heat energy moves through a flow system. Producing R-B convection involves isolating a liquid in a tiny enclosed cylindrical or rectangular cell and creating a temperature difference (gradient) between the bottom and top layers. It's the rough equivalent of heating a covered pan of water, though the experimental arrangement allows for precise control of fluid properties and the temperature gradient between the cylinder top and bottom, in addition to offering a safe view of the heated fluid's surface.

Under long-used experimental parameters, R-B experiments have consistently exhibited the same evolving structures, among them parallel rolls, squares and hexagons (discernible in the convection topography). The initial parameters and the type of fluid govern which kind of pattern will emerge first, but all the patterns are stable and non-chaotic. Given a constant temperature, they maintain their structure over time.

Changing Patterns

In 1992, Gunton's team replicated a laboratory experiment in which the expected transition from hexagonal to a parallel-roll state instead produced an unpredicted global spiral. The Gunton effort confirmed the experimental findings, revealing in a two-dimensional image a mesmerizing, stable rotating global spiral. "Convection often takes place in a very organized fluid motion," says Gunton, "and this has been studied and observed for decades. But supercomputing has helped reveal these novel, unpredicted states."

More recently, the Gunton team used the CRAY C90 to monitor the onset of Rayleigh-Benard from a non-convective state. The resulting two-dimensional images reveal a kaleidoscopic pattern of local spirals. Their findings matched those of an independent though simultaneous experimental effort, thus bolstering the veracity of the unpredicted find.

The spirals not only rotate, says Gunton, they move around in the fluid, annihilate each other, grow at the expense of one another and fluctuate in size and number. Gone is the regularity exhibited in the parallel rolls, with the fluid moving in continuous circular waves. Velocities and temperatures vary throughout the fluid. Some portions of it rise and fall faster than others. Everything about the system changes from one moment to the next. It's a crock pot of symmetry turned bubbly cauldron.

Chaos in Rayleigh-Benard Convection
The bull's eye-like symmetry of the global spiral (top) as compared to the kaleidoscopic nature of a collection of adjacent local spirals (bottom), as simulated on the CRAY C90 by James Gunton and coworkers. Color corresponds to temperature, increasing from violet to red.

"Since these local spirals are erratic in time -- changing position and size -- there's an irregular behavior in space as well," says Gunton. Thus, he says, a very simple experiment can now be used to study one of nature's most complex occurrences -- chaos. Chaos is a realm beyond order, whose disorder has a recognizable pattern over time. Chaotic patterns are seen in everything from weather systems to cardiac activity at the cellular level. "We're now using supercomputing to create three-dimensional models of the chaotic patterns," says Gunton, "which will provide further understanding of both convective behavior and chaos."

The new patterns revealed themselves when experimentalists, and later theorists, devised a means of enlarging the R-B cell, the enclosed setting in which the convection occurs and is observed. Previously, they could examine only a small piece of the convection system, which limited the extent to which pattern activity could evolve. "In order for us to mathematically model bigger cells, we had to solve some very complex equations," says Xi. "It couldn't have been done without supercomputing." The code ran between 300-350 Mflops on the C90 and the researchers have logged approximately 1200 hours on the supercomputers, with 900 more slated for investigations that will produce three-dimensional images of chaos in R-B convection.

"One of the quantities we try to calculate is the upward velocity of the fluid at different points in the system, at different times," says Gunton. "Solving the equations means determining the velocity of the fluid at any given point in the system at any given time. It can't be done without supercomputing."